4.1. Finite models

Note that models of K1–K3 are easy to come by on their own. The most flat-footed way to construct a model of these principles is to consider models of a higher-order form of nominalism: suppose that universal quantification into sentence position always yields vacuous truth because loosely speaking, there are no such things as propositions.Footnote ³¹ Then, K3 is vacuously true because it begins with a universal quantifier. As for K1 and K2, one can ensure their truth by having a possible world where is true for any interpretation of – this could be achieved simply by letting the extension of Q at this world include every set of worlds. So, for example, . (Recall that D(w) is not the domain of accessible worlds; it is the domain of existing propositions – in this case, no propositions exist at w.)

Thus, we get the following proposition:

Note that because this is a non-regular model, we do not get the principle which requires the domain to be non-empty. Indeed, one might want to go further and require not only that the propositional domains be non-empty but that they contain both and . Indeed, if the domain contains only these two propositions, we also get the desirable thesis that the existing propositions are closed under arbitrary Boolean operations.

With this mind, let us consider a slightly more interesting model of K3. Here, we let there be two worlds with a universal accessibility relation, such that at both worlds, the necessary and impossible propositions are the only propositions that exist: . Since there are only two existing propositions at either world, it is easy to find an interpretation of Q in which each of those propositions is the only proposition that is Q at some world: and . Thus, K3 is validated at both worlds in this model. It should be noted that the propositions that exist at each world satisfy a number of closure conditions: propositions corresponding to and exist, and an arbitrary Boolean combination of existing propositions exists. However, existence is not closed under the Q operator. For example, is true at the set of worlds , but does not exist even though the set of worlds corresponds to does.

What about K1 and K2? The set of worlds at which Prior's sentence, , is true is : at x a truth is Q, whereas at y, only the inconsistent proposition is Q. Since is not Q at any world, is not true at any world, and so is also false at both worlds. This precludes K1 being true, and thus also the schema K2 of which K1 is an instance.

We could validate K1 by changing our model slightly. Let , and as before, we stipulate that R is universal and moreover that only W and exist at each world, and that and . Finally, we stipulate that . Notice that K3 is valid in this model for the same reason as before. In this model, is true at y and z (it's true at y because only is Q at y, and vacuously true at z because nothing that exists is Q at z). Thus, expresses , and thus is true at z. Moreover, is vacuously satisfied at z, so the principle K1 – – is true at every world. Notice, however, that we can still find instances of K2 that are not validated. For example, has the singleton as its intension, which is not in the extension of Q at any world, so the instance of K2 is false at every world. Note as before that existence is closed under arbitrary Boolean operations, but not under the Q operator.

Let us see if we can push this strategy further and validate all three K principles. So far, we have only considered models in which one set of worlds appears in extension of Q at each world. But if we drop that constraint, we can generate a three-world model that validates K1–K3. Let , and let the extension of Q at x and y, the accessibility relation and the existing propositions be as before. Now let the extension of Q at z be . K3 is validated as above. K2 is validated because for any sentence that is true at some but not all worlds is such that (i) its intension is Q at z and (ii) it is vacuously true that every existing proposition at z that is Q is equivalent to it. Meanwhile, if is true at every world or none, then its extension will be Q without any existing proposition being Q at x and y, respectively.

This model is peculiar: it secures the validity of K2 by having the extension of Q be radically non-unique at z, even though the object language statement of uniqueness is satisfied at z. For example, at z, (i.e. the set of points ) is Q without any existing proposition being Q, and, likewise, (i.e. the set of points ) is Q without any existing proposition being Q.

The sense in which and its negation are uniquely Q is by no existing proposition being Q alongside them. What we want is to be able to impose a stronger uniqueness requirement; however, this turns out to be hard to express in the free logical setting. By analogy, suppose we wanted to say that someone is imagining Pegasus but not Atlantis or Zeus or.... The ‘...’ cannot be captured quantificationally because a quantificational claim won’t entail the instances involving fictional names.Footnote ³² However, we could introduce an operator for this purpose. The interpretation of ‘ snow is white’ is that it means the same as ‘snow is white is Q but grass is green is not Q, nor... ’ where the ellipsis is understood in this open-ended way. Two schemata partially characterising the behaviour of can be outlined:

These schemata must be understood open endedly.Footnote ³³ Model theoretically, we can interpret in our models as follows:

Armed with , there's a sort of uniqueness demand that strengthens K1–K3 that can be articulated as follows:(K1^+) (K2^+) (K3^+)

However, K1 fails in our present model; for example, is not true in the present three-world model. In fact, no three-world model will do.Footnote ³⁴ This idea generalises to show that no finite world model will do: there can be no finite model in which the set of worlds expressed by each sentence is the one and only member of |Q|(w) for some world w (and thus no finite model of K2 ). This holds even when restricting attention to the original language without .

4.2. Infinite models

It's natural to wonder if there are any infinite models satisfying the constraint. We can construct one in the spirit of the earlier models – involving a fairly frugal domain of propositions – as follows. Let , let the accessibility relation be universal and as before let . Now pick some bijection, , between and the set of finite and cofinite subsets of .Footnote ³⁵ Given , the extension of Q can specified as: .Footnote ³⁶ In this model, every finite and cofinite set is the unique member of the extension of Q at exactly one world. Moreover, it can be shown by a simple induction that in this particular construction, every closed sentence of our language expresses a finite or cofinite set.Footnote ³⁷ Thus, it follows that in this model, for every set of worlds expressed by some sentence, there is a world where that set is the unique member of the extension of Q. So we have a model of K1–K3, and also a model of K2 – and K1 and K3 . Just like in the earlier models, the propositions that exist at each world are closed under all Boolean operations (including infinitary conjunction and disjunction), but are not closed under the Q operator. For example, the set corresponding to exists, but is true at exactly one world, and no singleton set is in the domain of quantification at any world.

The models considered so far might be deemed uninteresting because they only permit quantification over the necessary and impossible proposition: there is both a paucity of propositions (there are only two) and a lack of variety (there are no contingent propositions). It's easy to modify the model so that it allows a few more propositions: for example, we could take a finite number of finite sets, and close them under Boolean operations, and the resulting model would still satisfy K1 –K3 . Of course, this model will only deliver finitely many existing propositions in total since closing finitely many finite sets under Boolean operations will only yield finitely many sets. As a corollary, existing propositions will not be closed under Q in these models: if expresses an existing proposition, and existence was closed under Q, then would express an existing proposition for each n. By the result above, any model of this sort that satisfies K2 or K3 will be such that expresses a different proposition for each n, so there must be infinitely many existing propositions.Footnote ³⁸ Since these models only have finitely many propositions, they therefore cannot simultaneously satisfy closure of existence under Q and either K2 or K3 .

It's natural to investigate models with an infinite variety of existing propositions, and in particular which satisfy the principle that existence is closed under Q. The preceding model can be adapted to this setting by expanding the domain to include all finite and cofinite sets. As in the earlier models, Qp always expresses a singleton of a world, if p expresses a finite or cofinite set. Thus, if p expresses an existing set of worlds, Qp expresses a finite and thus existing set of worlds. Notice also that existence is closed under the finitary Boolean operations, but not the infinitary Boolean operations: a finite combination of unions and complements of some combination of finite and cofinite sets will yield either a finite or a cofinite set, although infinite unions of finite sets might be neither finite nor cofinite.

Notice also that this is a model of K3 and K3 : every finite or cofinite set is the unique member of the extension of Q at some world.

However, the proof that every closed formula expresses a finite or cofinite set breaks down in this setting.Footnote ³⁹ It is no longer clear that this model validates K1 or K2. Indeed, we can show that the Prior sentence expresses an intension that is neither finite nor cofinite for any model of this sort.

Notice, of course, that by Proposition 4.3, any regular model of K3 will be infinite.

For the Prior sentence to correspond to a finite set, the propositions corresponding to each singleton set of worlds must be in nearly every case (i.e. in all but finitely many cases) be Q at the world contained by the singleton. (After all, in these models, only one intension is Q at each world, and so if there are infinitely many cases where a singleton is Q at a world that isn’t contained by the singleton, then there are infinitely many cases where every proposition that is Q at a world is false.) But that leaves only finitely many worlds to take care of the possibility of uniquely thinking each of the remaining finite and cofinite sets, which is impossible (since there are infinitely many that aren’t singletons). What about the hypothesis that the Prior sentence corresponds to a cofinite set? To rule this out, let us consider now not the singleton set of each world but its complement. For the Prior sentence to be true at cofinitely many worlds, it must be that in nearly all cases, the complement of a singleton be Q at the world contained in the singleton. (After all, if the complement of a singleton is Q at a world contained in that complement, then the proposition that is Q is true at the world in which it is thought and so will be false at such a world.) But if nearly all worlds are such that the complement of their singleton is Q at those worlds, then that leaves only finitely many worlds to take care of the possibility that each remaining finite and cofinite set is Q without anything else being Q, which is impossible.Footnote ⁴⁰ Recall now that it is a feature of our model that the only sets falling in the extension of Q are finite or cofinite sets since doesn’t express either a finite or a cofinite set, the extension of Q won’t include the set expressed by . Therefore, this model won’t satisfy K1, and ipso facto will not satisfy K2.Footnote ⁴¹

4.3. Rich models and ramification

In the above model, we’ve made space for a plenitude of existing contingent propositions, but offered no means for securing K2. Notice also that in the models of K3 given above, we have shown that the intension corresponding to any given sentence is the unique member of the extension of Q at some worlds. This doesn’t by itself guarantee that there is a possible world at which the proposition that snow is white and the proposition that grass is green are both Q without anything else being Q. A further constraint on the models is required if we want to ensure this and similar possibilities. Say that a model is rich if and only if, for any variable assignment, v, assigning variables to existing propositions, and set of formulae in the language, X, there is some world in the model at which the extension of Q is given by the set of intensions expressed by each member of X relative to v.

Here is a recipe for constructing a rich model that also ensures a plenitude of existing contingent propositions, while securing K1–K3 and K1 –K3 . In these kinds of models, we can think of the existing propositions as being arbitrary Boolean combinations of ‘basic propositions’ concerning only non-intensional matters. Following Kaplan, let's pretend that the basic propositions state facts about the distribution of earth, wind, fire and water (the details don’t matter much). We can model these basic propositions as sets of entities we shall call ‘proto-worlds’. We shall build up a richer space of possibilities in stages, and hence a space of intensions for formulae, with the proto-worlds corresponding to the possibilities at stage 1. Another layer of possibilities, roughly corresponding to which basic propositions are Q, can be generated by considering ordered pairs of proto-worlds with sets of sets of proto-worlds, where the latter component corresponds to the stage 1 intensions that are Q at that possibility. Call these stage 2 possibilities. Now there are more sets of stage 2 possibilities that can be Q than merely sets of proto-worlds, so we must repeat the process. We can generate a third layer – the stage 3 possibilities – by considering ordered triples consisting of a proto-world, a set of sets of stage 1 possibilities (proto-worlds) and a set of sets of stage 2 possibilities. This iterates in the obvious way, with our complete possibilities (or ‘worlds’) being infinite ( -length) sequences where the th component of the sequence is sets of sets of stage n possibilities, giving us the stage n intensions that are Q at that stage.

For a technical reason, that will become apparent later, we also need to stipulate that at each stage, we only add sets of intensions that are ‘new’ – intensions that couldn’t have been generated at an earlier stage. A set of n-tuples X is generated by a set of m-tuples Y for iff X is the set of all n-tuples that extend m-tuples in Y; thus, we will require that the th component of our infinite sequences consist only of sets of n-tuples that are not generated by a set of m-tuples for smaller m.

Our complete intensions are just sets of these infinitary sequences. Notice that our possibilities, as we have constructed them, do not directly tell us which sets of complete intensions are Q at each possibility: they give us an infinite sequence of incomplete (i.e. finite stage) intensions, none of which belong to our space of complete intensions. However, each stage n intension can be extended to a complete extension in a unique way: if I is a stage n intension, we can consider the set of all infinite sequences in our space, , that extend the elements of I. (Intuitively, we can think of as an intension that settles some questions about which stage n propositions are Q, but leaving it completely open what higher stage intensions are Q.) Let us call the extension of a stage n intension to the space of complete intensions a completed stage n intension. Finally, call an intension bounded if and only if it is a completed stage n intension for some finite n.

We are now in a position to construct a model. Given any set of proto-worlds, we can construct a set of worlds W consisting of infinitary sequences as described above. As usual, we will let our accessibility relation be the universal relation. The domain of the propositional quantifiers, D(w), will be the same at every world: the set of all completions of sets of proto-worlds, i.e. the set of all completed stage 1 intensions. The extension of Q at a world w, will contain, for each stage n intension belonging to the nth component of w, the completion of that stage n intension: . So the extension of Q is a set of sets of infinitary sequences, each such set being a completed stage intension that encodes what is Q at a certain particular stage.

Two things are worth noting about this construction. Firstly, the completed stage n intensions are closed under arbitrary Boolean operations. This means in particular that the propositions that exist at each world – completed stage 1 intensions – are closed under arbitrary Boolean operations. Secondly, if expresses a completed stage n intension, and is not a completed stage m intension for any , then will express a completed stage intension.Footnote ⁴² Thus, in particular, the completed stage 1 intensions are not closed under the Q operator.

This model validates K1–K3 and K1 –K3 . We’ll show this by demonstrating two things: (i) for every bounded intension, there is some world where that intension is uniquely in the extension of Q and (ii) every closed formula expresses a bounded intension. For (i), we can in fact show something stronger: for any set of bounded intensions, there is some world where they are the only intensions in the extension of Q. This fact will guarantee that our model is ‘rich’ in the sense defined earlier. Given a set X of bounded intensions, we can construct a world in the following way: pick any proto-world for the first component, and then for the nth component of the sequence simply take the set of n stage intensions whose completions are in X. The proof of (ii) is a straightforward induction on the complexity of formulae, which we leave to a footnote.Footnote ⁴³

Since the propositional quantifiers range over completed first stage intensions, and thus bounded intensions, (i) guarantees that K3 and K3 are validated. Moreover, given (ii), every closed sentence of the language expresses a bounded intension, so K2 and K2 (and thus K1 and K1 ) are also validated.

As noted above, the domain of the propositional quantifiers is closed under arbitrary Boolean operations at each world, but not under the Q operation. It is possible to tweak the above model so that propositional existence is closed under the Q operation, is closed under finitary Boolean operations but is not in general closed under infinitary Boolean operations. This is achieved by iterating the construction to instead of , and letting the propositional quantifiers range over (completions of) -bounded intensions, i.e. intensions that appear at some finite stage. In this model, one can show that (i) every -bounded intension (i.e. any intension appearing at a countable stage) is uniquely in the extension of Q at some world and (ii) every closed formula expresses an -bounded intension, guaranteeing the validity of K1–K3 and K1 –K3 . Since the details of this construction are more involved, we omit it.

The first model we considered above is interesting for another reason: it naturally accommodates the addition of a hierarchy of propositional quantifiers roughly in line with a broadly Russellian approach to higher-order logic. Indeed, for each n, we can introduce quantifiers into the language that range over the completed n-stage intensions. Such quantifiers will satisfy a restricted version of universal instantiation, guaranteeing that the higher quantifiers range over larger domains of propositions:(UI_n)

where Q embeds within itself at most n times in .

In these models, we also obtain a generalisation of K3 and K3 for each propositional quantifier, for example:(K3_n^+)

The resulting models bear a family resemblance to Kaplan's description of a construction designed to accommodate K3 . We’ll discuss these connections later in Section 7.1.

Augmenting our models in this way thus gives us the following result:

6.1. Paradoxes from closure assumptions

In the models above, we noted that the domain of the propositional quantifiers was often closed under some operations but not others. It is natural to wonder whether this is an accident, or if there is an obstacle to satisfying all of the closure principles that we gestured at.

To sharpen matters, let's look at what these closure conditions amount to in the object language. Consider the following closure principles:(CL \top) (CL\bot) (CL\lnot) (CL\Box) (CL{ Q}) (CL\wedge) (CL\equiv) (CL\forall)

where does not contain q or r free.

In all but the last case, the CL principle corresponds to closure of the domain under the model theoretic operation corresponding to the connective . In the case of CL , closure under arbitrary conjunctions suffices for its truth, although there is no condition statable in the object language under consideration that precisely corresponds to the closure of the domain under arbitrary conjunctions.

Each set of connectives, X, the above principles corresponding to X along with the principle (a principle ensuring that there's at least one proposition) and the normal axioms governing proposition identity give rise to a corresponding restricted version of universal instantiation:UI({ X})

where only contains connectives in X.

None of our models validated all of these principles. There is a fairly systematic reason for this.

Informally, if one has all of the closure principles, one can prove the existence of the Prior proposition from which one can derive a contradiction, given K1 or K3 and other background principles. Given CL , CLQ and CL , it is straightforward to show (noting that is definable from and ). An instance of CL states that . Then, given the existence of at least one proposition (as guaranteed by CL ), we can infer from these two things that . Given that the Prior proposition exists, and the usual axioms governing propositional identity, one can prove a contradiction from K1. Another way to see this is that in the proof of Prior's theorem, we only appealed to an instance of UI(), and given CL , we can prove .

Note here that the fact that we are using identity rather than necessary equivalence in the statements of our closure principles plays an important role in this derivation. It could be that for any Boolean combination of existing propositions, there is an existing proposition necessarily equivalent to it. In particular, there could be a proposition necessarily equivalent to the Prior proposition. But unless this existing proposition is identical to the Prior proposition, we cannot infer that I’m thinking something from the fact that I’m thinking the Prior proposition.

Closure principles are special instances of general principles, telling one how to combine elements from various types to form elements of other types (another example is the principle that for every x and every F, there's a p identical to Fa: .) It's a feature of the free logical approach that it is forced to deny a variety of principles of this sort at the type of propositions that are prima facie elegant. Of course K1–K3 are prima facie compelling too, but one wonders whether the gain outweighs the cost.

We should note in passing that there are a number of different kinds of pictures that may be used to flesh out the free logical approach in rather different ways. One kind of constraint is suggested by Russell's vicious circle principle, applied to the propositional type: the constraint that a proposition cannot quantify over a domain that includes itself. But a rather different constraint is suggested by the idea that operators corresponding to intentional attitudes can often take one from the existent to the nonexistent. The first constraint is naturally aligned with the rejection of CL . Suppose that for every p, the proposition that exists, then CL entails that which violates the vicious circle principle: is in the range of , and thus also in the range of , meaning that quantifies over a domain that includes itself.Footnote ⁴⁸ The second constraint naturally leads one to reject CLQ: when Q corresponds to an intentional attitude, the result of combining Q with an existing proposition may take one to the nonexistent. Of course within this approach, there are all sorts of further decision points. One might think that any sentence containing ‘Q’ fails to express an existing proposition, or one might instead think that only the paradoxical ones fail to do so. (If one takes the latter approach, one might also very well deny CL since there are sentences for which no paradox attends the assumption that it expresses an existent proposition, but from whose negation a paradox does follow from the correlative existence assumption.) Moreover, the latter approach naturally suggests higher-order contingentism. It will, for example, be a contingent matter whether assuming the existence of the proposition that the guy sitting in room 101 at time t is thinking only falsehoods leads to paradox. A more necessitist friendly version of the second idea will say that any sentence that could have led to trouble in fact doesn’t express an existing proposition.

6.2. Comprehension and supervenience

Whatever one's motivations are for restricting the realm of the existing propositions, it is natural to assume that there remains a realm of fundamental propositions whose existence is unthreatened by intensional paradoxes, or by vicious circle constraints. These may be facts about the field values at particular space–time points, but in the current context, no harm will be done to once again assume the imagery of a crude version of pre-Socratic physics, in which the fundamental consists facts about earth, air, fire and water. Further, it appears harmless to assume that arbitrary conjunctions and negations of such propositions similarly exist, for they only concern fundamental matters, and moreover harmless to assume that these propositions exist necessarily if they exist at all.

It is further a metaphysically compelling thought that all truths supervene on these fundamental propositions: that no two possibilities can agree with regard to the truth of all earth, air, fire and water facts but disgree about some intensional matter – e.g. whether I’m thinking something, or about whether everything I’m thinking is false. Once supervenience is assumed, then given our Boolean liberalism regarding the fundamental propositions, it is a short step to saying that every sentence is necessarily equivalent to some fundamental proposition. This in turn entails, given the existence of the fundamental propositions, that every sentence is necessarily equivalent to an existing proposition:

This is, of course, just the comprehension principle Comp we introduced is Section 3.

If necessary equivalence guarantees propositional identity, then Comp is tantamount to saying that every sentence expresses an existing proposition, which in turn prevents all the free logical manoeuvres for saving K1–K3. But might we move to a hyperintensional setting in which Comp is admitted for supervenience theoretic reasons, but in which K1–K3 are saved by a combination of hyperintensionality and free logic? For example, even given the modal equivalence of some existing proposition, call it ‘Fire’, with the sentence ‘everything I’m thinking is false’, our original paradox can be blocked. Given hyperintensionalism and free logic, it's coherent to suppose (i) that I’m thinking that everything I’m thinking is false, (ii) that there's some proposition (i.e. Fire) necessarily equivalent to its being the case that everything I’m thinking is false and yet (iii) there's no proposition that I’m thinking. Now this is indeed a way of saving K1 while accepting Comp. For example, consider a two-world model, . Note that since the accessibility relation is the identity relation, just amounts to truth, and Comp just amounts to the existence of a true and a false proposition, which is satisfied at both worlds (similarly, propositions exist necessarily for the same reason, even though we have a variable domain model). At both worlds, no existing proposition is Q (W does not exist at either world), even though there are existing intensions necessarily equivalent to the nonexistent intension at each world ( exists and is necessarily equivalent to W at x, similarly for y and ). Since nothing is Q at either world, then is vacuously true at both worlds, thus has the intension W. Since this is Q at both worlds, K1 (and K1 ) is true at both worlds. Indeed, one can construct (infinite) hyperintensional models that validate K2 and K2 , at least, for the language fragment under consideration.Footnote ⁴⁹ We can summarise our results with the following proposition:

But it is instructive to see that even in this hyperintensional setting, K3 cannot be salvaged by free logical resources without giving up the assumptions made at the outset. Assume as before that Fire, being a fundamental proposition, exists necessarily, and that it is necessarily equivalent to its being the case that everything I’m thinking is false. K3 requires that it be possible that Fire is thought and that anything else thought has the same truth value as Fire. But whether or not Fire is identical to the proposition that everything that's thought is false, this requirement of K3 is paradoxical. For if Fire is true at the world it is thought, then since Fire is sufficient for it being the case that everything thought is false, then Fire is also false at the world it is thought. And if Fire is false at the world it is thought, and everything thought at that world shares the same truth value as Fire, then everything thought at that world is false, in which case, Fire cannot after all be modally equivalent to it being the case that everything thought is false. So in the current setting, we have to concede that Fire is a counterexample to K3 and cannot be thought without thinking a proposition with a different truth value. More formally, we’ve sketched an argument for the following proposition:

Note that our assumption strengthens Comp by assuming that the necessarily equivalent proposition to exists necessarily, which is in line with the thought that fundamental propositions like Fire exist of necessity.

With this concession made, the benefits of free logic seem fairly scant. There are a few ways to resist this kind of argument against K3, but all seem a little desperate. One is to deny that arbitrary Boolean combinations of fundamental propositions preserve existence: that is to say, a conjunction, disjunction or negation of earth, fire and water facts might not exist. It could be that every truth is fixed by the distribution of truth values over existing fundamental propositions, but if conjunctions and disjunctions of these propositions needn’t exist, it would be unreasonable to suppose that each truth is modally equivalent to any particular existing proposition. Another move would be to deny that the fundamental propositions exist necessarily. For example, supposing Fire exists and is necessarily equivalent to it being the case that everything thought is false. Perhaps at worlds where I think that everything I think is false and nothing else, Fire doesn’t exist.

6.3. Paradox without universal instantiation

Free logic provides a way of keeping K1–K3, but K1–K3 are just instances of a wider pattern of intuitive judgements, many of which cannot be saved by the free logical manoeuvres we’ve been exploring. For example, while K1 guarantees that it's possible to think and only thing materially equivalent to , one might think it is similarly possible to think all and only the propositions that are materially equivalent to . We could formalise this idea as follows:(K4)

Of course, if many propositions exist, this requires that one think a lot of propositions since K4 requires that at the witnessing world, every proposition or its negation is thought. But it's tempting to think that K4 represents a metaphysical possibility (and this would be particularly plausible if one adopts a lax interpretation of thinking in which it is closed under logical consequences.) However, K4 is inconsistent with the assumption that a truth and a falsehood exist – which would be guaranteed if we accepted the minimal principles CL and CL .Footnote ⁵⁰ Notice moreover that we can make similar trouble for the weaker hypothesis that one thinks some proposition materially equivalent to and no proposition that is not materially equivalent to :(K5)

If K5 is true, it is because it is possible to think an (existing) truth and only truths or an (existing) falsehood and only falsehoods. To think an existing truth would mean that it's false that everything thought is false, i.e. is false, in which case, one wouldn’t have succeeded in thinking only existing propositions materially equivalent to . Meanwhile, if one thought only existing falsehoods, would be true, and we would not have succeeded in thinking an existing proposition materially equivalent to (i.e. an existing truth). This informal argument can be turned into a rigorous proof in propositionally quantified free logic. A closely parallel argument can be run against K4 and the additional assumption of the existence of a truth and a falsehood (as guaranteed, if we accept CL and CL ).

A related observation is that assuming the existence of a truth and a falsehood, there will be interpretations of Q for which a version of Prior's theorem is still provable. Let's interpret Q to mean ‘think ’, where one thinks a proposition so long as one thinks a proposition materially equivalent to it. Then, one can prove that if one thinks that everything one thinks is false, then one thinks something true and thinks something false.

It is worth taking a closer look at the Russellian treatment of these paradoxes. Supposing we read the quantifier as the Russellian's zero-level quantifier , so that the quantifiers appearing in and the wider scope quantifiers appearing in K4 and K5 are both of zero level. As it stands, neither principle violates the vicious circle principle: even though may express a first-level proposition, it's still true that there are zero-level propositions materially equivalent to it, and thus that one can divide the zero-level propositions into those materially equivalent to it and those not materially equivalent to it.Footnote ⁵¹ Given these observations, K4 and K5 – interpreted in terms of zero-level quantification – should look just as appealing to the ramifier as anyone else; yet, they are plainly inconsistent.

Notice that a proponent of ramification might attempt to retreat to a weaker form of K4 and K5 in which the outer quantifiers range over the same level as the level of the proposition expresses (which will have to be higher than zero because contains zero-level propositional quantifiers). Thus, for example, we might weaken K4 to the following principle:(K4^-)

Unfortunately, this principle is also problematic. Assume, as customary, that every zero-level proposition is a first-level proposition (that is, the levels are ‘cumulative’), and assume also that there is at least one zero-level truth, and at least one zero-level falsehood. Consider the world that witnesses the truth of K4 . Either I’m thinking a zero-level truth or I’m not at this world. If I am, then is not the case; thus, if I’m only thinking first-level propositions materially equivalent to , I’m only thinking first-level falsehoods. But by assumption, I’m thinking a zero-level truth, and thus a first-level truth since every zero-level proposition is a first-level proposition. On the other hand, suppose that I’m not thinking any zero-level truths. This gives us the truth of , which means, since I’m thinking all and only the first-level propositions materially equivalent to , that I’m thinking every first-level truth, including the zero-level truth, contradicting our assumption.

Given all of these observations, one might wonder just how much we have really achieved by moving to higher-order free logic. The proponent must find a principled background picture that motivates K1 (say) without also motivating the outright inconsistent principles K4 and K5. Here is one way you might attempt to flesh out that picture: one could argue that thinking (and nothing else) is an easy matter. All one needs to do to achieve that is to mentally token something corresponding to the sentence ‘everything I’m thinking is false’, without tokening anything else at the same time. By contrast, it is not so easy to ensure the existence of a proposition by performing these mental tokenings – the existence of propositions is beyond your mental powers. This picture naturally motivates K1 (and indeed K2 if we generalise to other sentences), but note that it does not motivate K5 since K5 requires it to be possible for you to think something existing materially equivalent to – tokening does not guarantee the existence of anything. Similarly, it's unclear what you could mentally token in order to ensure the truth of K4: since it is beyond your control which propositions exist, merely tokening a number of sentences does nothing to guarantee that you think all and only the existing propositions materially equivalent to .

Although one can in principle outline a picture in which K1 and K2 are motivated without also motivating K4 and K5, we feel that once one has stopped taking these kinds of compelling judgements at face value, one wonders whether we shouldn’t just accept the original consequences of UI for K1 and K2.

6.4. Expressive limitations

Given UI, one can in a higher-order framework make extremely general claims by recourse to higher-order quantification. For example, if we think that knowledge is factive, one can capture this idea quite straightforwardly with the following sentence:

Similarly, one can state that all conjunctions have true conjuncts in a way that is not limited to the conjunctions expressable in any particular language:

Now even when we relinquish UI, we may write down these things, and remain confident in them. But in a free logical setting, they may feel insufficiently general. For example, supposing there is no such proposition as , we may feel the urge to write down generalisations for which the following are special cases:

However, the quantified sentences specified above are unfit for this purpose because they do not entail these instances.

A natural thing to try to go for is a schema rather than a quantified claim. For example, the factivity of knowledge could be captured by the following schema:

An initial concern with this strategy is that it won’t satisfy our desire to write down a general truth since schemas are neither true nor false (even though they can have true or false instances).Footnote ⁵² Note that this cannot be readily circumvented by instead writing down a metalinguistic generalisation to the effect that all uniform interpretations of the schematic letters deliver truths since this reintroduces propositional quantification and hence may not have the requisite generality.

In order to avoid quantification over interpretations of sentence letters, one might opt for a metalinguistic generalisation that is substitutional in nature, along the following lines: every uniform substitution of a schematic sentence letter by a meaningful sentence in the language yields a truth. But this would lose generality for a different reason since, for example, the generalisation about factivity is supposed to cover cases of knowledge that aren’t expressible in our language. A similar concern would attach to any effort to express the required generalisations using infinitary conjunctions or disjunctions of sentences in our finitary language. Note that arguably no infinite conjunction can contain all sentences of the infinitary language as conjuncts, for then it would contain itself as a conjunct and thus be non-well founded.

Thirdly, quite aside from the desire to express bona fide truths, the schematic approach is particularly problematic in certain embedded environments. Take the generalisation that in the classical setting, we would happily express with the following formula:

That is, it is possible that every proposition is Q. Now suppose we felt this inadequate because we wished to convey the thought that it's possible to think, alongside all the existing propositions, , and and so on, whether or not the latter corresponds to existing propositions. In this connection, the following schema will not be of any use:

After all, that schema does not require that and be thought together in any single world. This point generalises to occurrences of universal quantifiers in embedded environments where, in the classical setting, the embedded universal quantifier wouldn’t commute with the embedding.

One might hope to bypass a number of these concerns by appeal to a device alluded to earlier. One might try to express some of these generalisations using ellipsis in the following kind of way:

Note that for this to do the requisite work the ‘...’ cannot merely be a stand in for an infinitary conjunction of sentences in the finitary language, for this lacks the generality we were seeking. The ellipsis must, for example, be understood at least to cover all possible extensions of the language. So let us assume that the object language has been extended to include exactly this kind of ellipsis.

There are a number of pressing concerns here. One is that the more careful one is about the syntax of this device, the more it begins to resemble a quantifier. For example, one will have to introduce a variety of ‘...'s corresponding to the need for different propositional variables in the original setting. But we’ll focus on a particular dilemma. Are ‘...’ thoughts to be understood so as to be under the scope of ‘...’ generalisations? For example, is the claim that if I know that grass in green, then grass is green and... to include as a special case that if I know that [if I know that grass is green then grass is green and...], then [if I know that grass is green then grass is green and...]? If the ellipsis is indeed a legitimate extension of the language, and ‘...’ claims are supposed to cover all legitimate extensions of the language, then ‘...’ claims had better be special instances of ‘...’ claims. An initial worry is that this raises the spectre of non-well foundedness. But more importantly, the ‘...’ claim seems now to behave enough like a classical quantifier as to allow one to reinstate the Priorian paradoxes. Consider K1 : Once one has gotten the hang of ellipsis, K1 should look just as compelling as the original K1. Note that from one can conclude for any in our language (whether includes ellipsis or not). This principle is exactly analogous to the principle of universal instantiation that caused problems for K1 in the first place.

6.5. First-order free logic

We’ve have in effect been developing pessimistic pictures of the prospects for a higher-order free logic that's motivated by the prospect of keeping principles like K1–K3. It's less clear whether the materials of this section can be developed into a case against higher-order free logical approaches that are motivated by other concerns (for example, by the desire to be a contingentist about the existence of properties). Moreover, it's even less clear whether these considerations carry over to the first-order setting. For example, concerns about closure under Boolean operations have no analogue in the domain of individuals nor do concerns about supervenience. Of course expressivity worries, and in particular the challenge of finding suitable ways of formulating perfectly general principles, are likely to be a common theme among those who oppose free logical approaches in general.

We are not going to systematically explore the resources of various free logicians for enhancing expressive power. But we will mention one theme that makes things particularly difficult for one who promotes free logic as a means for saving K1–K3. Suppose, for example, one is a first-order free logician. One might still be attracted to the schema , for that allows one to use a modalised quantifier to formulate perfectly general claims, for which singular terms are instances. Similarly, one might pursue such a strategy for a higher-order free logic. However, such a strategy is dangerous, to say the least, if ones motivation for higher-order free logic is the preservation of K1–K3: . This is because modalised quantifiers, once accepted, may well come to form the basis for revenge-like versions of Prior's paradox.

7.1. Ramification and Prior–Kaplan puzzles

We’ve distinguished some varieties of ramification, but how do these varieties interact with K1–K3? Let's examine in turn first the strategy of ramifying the attitudinal operators and then the strategy of quantificational ramification. We’ll then examine whether there are special advantages to ramifying along both fronts when it comes to the Prior–Kaplan puzzles.

Suppose first one leaves higher-order quantification unramified, but does ramify operators like ‘it is thought that’.Footnote ⁶³ We’ll focus here on the standard kind of ramifier that prohibits the truth of any sentence that has an attitudinal operator appear within either its own scope, or the scope of an operator of lower type (as noted, there is a further divide among those who think that such occurrences are either false or ungrammatical).Footnote ⁶⁴ Obviously, this framework won’t validate K1 on any uniform interpretation of ‘Q’, for Q appears within its own scope in K1 (and so it is either false or ungrammatical). Generally, when one writes down a schematic principle, one intends the schematic variables to be interpreted uniformly. As applied to K1, it seems as though we shouldn’t have been writing it down after all. That said, the present kind of ramifier can offer a diagnosis of why claims like K1 seem attractive. It's natural for the proponent of this view to contend that when, for example, in ordinary thought and talk we say things like ‘I thought Bob thought it was Tuesday’, we don’t automatically say something false or incoherent. Rather, without properly keeping track of what we’re doing, the two occurrences of ‘thought’ correspond to two things at the level of logical form. From this perspective, there's a kind of logical form ‘blindness’ that ordinary people are susceptible to, and which could explain why they would mistake K1 for a family of well-typed principles that could be true. Indeed, it's easy to find models in which one can uniquely think the proposition that everything I think is false.Footnote ⁶⁵

This style of explanation can’t be easily extended to the puzzles associated with K3, or the variant Kaplan discusses, K3 : . Both principles are well-typed by the lights of the ramifier. Moreover, Kaplan's original considerations telling against K3 seem to be just as compelling in this setting. There are still many more propositions than worlds, so not every proposition can be uniquely thought – the fact that there will be a whole hierarchy of additional unsatisfiable principles is of little comfort. As for K3, one cannot run Prior's original argument because certain steps in the proof violate the typing constraints. However, Prior's original argument can be adapted so as to respect those typing constraints, e.g. by saying things like ‘there's something identical to the proposition that everything I’m thinking is false, and I could have thought it and only propositions materially equivalent to it’. One could block this trick by taking seriously the idea that the quantifiers are typed, and thus that the variable flanking the right-hand side of the identity connective cannot appear under the scope of ‘thinks ’ since it must be of the same level as the sentence on the left-hand side (‘everything I’m thinking is false’ which is of level 2). But this would no longer be the selective kind of ramification under consideration.Footnote ⁶⁶

What about K2? A natural thing for the ramifier to try is to say that K2 is true for any well-typed instance.Footnote ⁶⁷ A modification of the model construction in proposition 4.7 from Section 4.3 serves to show this is consistent.Footnote ⁶⁸ On this way of understanding K2, K1 isn’t an instance of it since it violates the typing constraints. In this setting, the supervenience problems discussed in Section 6.2 are still relevant: supposing that thinking facts supervene on earth, fire, wind and water facts, then for all we know, there's an earth, fire, wind and water sentence that expresses a proposition that's necessary and sufficient for thinking that everything I’m thinking is false (of course, supervenience doesn’t guarantee that there is such a sentence). The instance of K2 with this sentence would not violate typing constraints, but as we saw in Section 6.2, this instance would be false. That said, one might, taking a leaf from Kaplan book, have the more modest ambition of saving the logical coherence of K2, rather than the truth, and in particular, its compatibility with various plausible metaphysical hypotheses.Footnote ⁶⁹

Let us turn to the ramifier who selectively ramifies quantification, but does not ramify the attitudinal operators. There are issues of detail concerning whether it is the quantifiers, variables or lambda operator that are to be ramified here; we won’t fuss about this, but for simplicity, we will work with a picture in which it is the variables that get ramified and that determiners like ‘all’ are no more typed than logical operations such as ‘not’. (Moreover, while the abstract idea of typing quantifiers does not force any kind of cumulative hierarchy, we’ll imagine, with the standard ramifier, that the domains of these quantifiers are linearly ordered by inclusion.)Footnote ⁷⁰

By contrast with the last strategy discussed, this kind of ramifier would with Kaplan have legitimate concerns about possible world semantics as it is standardly understood. Possible world semantics encourages one to think that from within one's language can quantify ‘all at once’ over a complete and unextendable domain of propositions. But for this kind of ramifier, this is an illusion: each quantifier ranges over a domain of propositions, each of which may have maximally strong elements. For each level of quantifier, these maximally strong elements become stronger and stronger with no end – while possible worlds may be the limit of these elements, there is no way of simulating quantification over these limits.Footnote ⁷¹

What happens to K1–K3 on this interpretation? As noted in Section 1, if the different levels of quantification are not to collapse into one another, then each quantifier must abide by a free logic (this is due to Harris's theorem).Footnote ⁷² What bears emphasis is the tight connection between the ramified quantifier picture and the discussion of free logic in the earlier sections of this paper. For note that on any interpretation of the quantifiers, universal instantiation – and coordinately existential generalisation – will fail in just the kind of way discussed in those sections. Thus, if I’m saying that everything I’m saying is false, there's nothing (although something ) that I’m saying; and similar kinds of results can be produced for the quantifier ‘something ’ and so on. So each quantifier is governed by a positive free logic. Note that in those sections, as here, we are not typing ‘thinks’ in such a way that prohibits it appearing within its own scope.

It is worth noting that in Section 4.3, we provide a model that explicitly accommodates a hierarchy of quantifiers of the sort envisaged here (proposition 4.9). And in that context, we provided a model which satisfies K2 (on any interpretation of its quantifiers), and even the stronger K2 . Note that these schemata now include instances that involve the new typed quantifiers that we have introduced to the language in this environment. It thus follows that for any interpretation of the quantifiers in K1 that version of K1 will be an instance of our schema K2 (and a similar relation holds between K1 and K2 ). As for K3 and K3 , there will be a different version of the principle for each quantifier, which is also validated in the model described in the proof of proposition 4.9.

Given these results, it is clear enough that as far as Prior–Kaplan puzzles are concerned, there is no immediate benefit to ramifying both the quantifiers and the attitudes. Indeed, the dual ramification strategy would seem to have significant costs in this setting, viz K1–K3, after all if one merely ramifies quantifiers, one can easily write down K1–K3 allowing that it is governed by the standard convention that ‘thinks’ has a uniform interpretation. But once one has ramified the attitudes, an endorsement of K1 appears much more problematic. Of course there may be other philosophical motivations for a more thorough going ramification. However, considerations pertaining to the Prior–Kaplan paradoxes seem very weak in this regard.